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409Hw06ans - STAT 409 Fall 2011 Homework#6 due Friday...

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STAT 409 Homework #6 Fall 2011 ( due Friday, October 14, by 4:00 p.m. ) 1. In the past, only 30% of the people in a large city felt that its mass transit system is adequate. After some changes to the mass transit system were made, we wish to test if the proportion of individuals who feel the mass transit system is adequate has increased, that is, test H 0 : p = 0.30 vs. H 1 : p > 0.30 . Let X denote the number of those who feel the mass transit system is adequate in a random sample of 20 persons. H 0 : p 0.30 vs. H 1 : p > 0.30, Right – tail. a) Suppose we decided to use the rejection region “Reject H 0 if X 11.” Find the significance level α associated with this rejection region. α = P ( Type I error ) = P ( Reject H 0 | H 0 true ) = P ( X 11 | p = 0.30 ) = 1 – CDF ( 10 | p = 0.30 ) = 1 – 0.983 = 0.017 . b) Find the Rejection Rule with the probability of Type I Error closest to 5%. Decision rule: Reject H 0 if X b . Want P ( Type I error ) = 0.05. P ( Type I error ) = P ( Reject H 0 | H 0 true ) = P ( X b | p = 0.30 ) = 1 – CDF ( b – 1 | p = 0.30 ). Want ( 1 – CDF ( b – 1 | p = 0.30 ) ) 0.05, CDF ( b – 1 | p = 0.30 ) 0.95. CDF ( 9 | p = 0.30 ) = 0.952, b – 1 = 9, b = 10. Decision rule: Reject H 0 if X 10 . c) What is the actual value of the probability of Type I Error for the Rejection Rule in part (b) ? α = P ( Type I error ) = P ( Reject H 0 | H 0 true ) = P ( X 10 | p = 0.30 ) = 1 – CDF ( 9 | p = 0.30 ) = 1 – 0.952 = 0.048 .
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d) Suppose that 8 persons out of 20 feel the mass transit system is adequate. Find the p-value. p-value = P ( value of X as extreme or more extreme than X = 8 | H 0 true ) = P ( X 8 | p = 0.30 ) = 1 – CDF ( 7 | p = 0.30 ) = 1 – 0.772 = 0.228 . 2. A state legislature says that it is going to decrease its funding of a state university because, according to its sources, 40% of the university’s graduates move out of the state within three years of graduation. In an attempt to save the university’s funding, you want to show that the proportion of graduates who move out of state is less than 0.40, and decide to test H 0 : p = 0.40 vs. H 1 : p < 0.40 . Let X denote the number of graduates in the sample of 20 graduates who move out of state within three years of graduation. H 0 : p = 0.40 vs. H 1 : p < 0.40. Left-tail. a) Suppose we decided to use the rejection region “Reject H 0 if X 3.” Find the significance level α associated with this rejection region. α = P ( Type I error ) = P ( Reject H 0 | H 0 true ) = P ( X 3 | p = 0.40 ) = CDF ( 3 | p = 0.40 ) = 0.016 .
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b) Find the best Rejection Rule with the significance level α closest to 0.05 . Decision rule: Reject H 0 if X a . Want P ( Type I error ) = 0.05. P ( Type I error ) = P ( Reject H 0 | H 0 true ) = P ( X a | p = 0.40 ) = CDF ( a | p = 0.40 ). Want CDF ( a | p = 0.40 ) 0.05. CDF ( 4 | p = 0.40 ) = 0.051, a = 4. Decision rule: Reject H 0 if X 4 . c) What is the actual value of the significance level for the Rejection Rule in part (b) ? α = P ( Type I error ) = P ( Reject H 0 | H 0 true ) = P ( X 4 | p = 0.40 ) = CDF ( 4 | p = 0.40 ) = 0.051 . d) Suppose that 6 out of 20 graduates in your sample moved out of state within three years of graduation. Compute the p-value of the test.
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